2026-09-21T20:48:29Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/129887 2025-10-25 col_2142_5131 col_2142_8888 com_2142_5130 com_2142_8887 com_2142_234

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-20 without embargo terms The student, Tanya Veeravalli, accepted the attached license on 2025-07-17 at 13:05. The student, Tanya Veeravalli, submitted this Dissertation for approval on 2025-07-17 at 13:13. This Dissertation was approved for publication on 2025-07-18 at 15:30. DSpace SAF Submission Ingestion Package generated from Vireo submission #22616 on 2025-10-20 at 16:58:41 Geometric and functional representations of stochastic neural dynamical systems: from realization theory to controlled approximation Veeravalli, Tanya 2025-07-18 Raginsky, Maxim Raginsky, Maxim Srikant, Rayadurgam Belabbas, Mohamed Ali Zhao, Zhizhen Optimal Control Neural Dynamical Systems Stochastic Realization Theory Nonlinear Controllability Function Approximation eng There has been a great deal of interest in understanding continuous-time processes in deep learning, in particular methods related to (stochastic) control for improving diffusion models. In this work, we explore various facets of function approximation and realization problems through the lens of dynamical systems theory, neural stochastic differential equations (neural SDEs), and differential geometry. A neural SDE is an Itô diffusion process whose drift and diffusion matrices are elements of some parametric families. We cover topics from estimating the transition density of both uniformly elliptic and possibly degenerate diffusion processes by leveraging tools from sub-Riemannian geometry and stochastic control. There are many nuanced insights we can get into the behavior of deep neural networks and diffusion models by studying properties of associated problems in optimal control theory and drawing on other tools from the rich mathematical physics literature. The geometric insights explain the underlying noise structure and controllability properties of a stochastic dynamical system while also explaining the expressive power of the stochastic system. 2025-08 Text https://hdl.handle.net/2142/129887 Copyright 2025 Tanya Veeravalli Electrical & Computer Eng Electrical & Computer Engr University of Illinois Urbana-Champaign Ph.D. Dissertation